# ACT Math : How to find the endpoints of a line segment

## Example Questions

### Example Question #1 : Midpoint Formula

In the standard (x,y) coordinate plane, the midpoint of  line XY is (12, 3) and point X is located at (3, 4). What are the coordinates of point Y?

(21, 10)

(9, 7)

(7.5, 0.5)

(9, 7)

(4, 11)

(21, 10)

Explanation:

To get from the midpoint of (12, 3) to point (3,4), we travel 9 units in the x-direction and 7 units in the y-direction. To find the other point, we travel the same magnitude in the opposite direction from the midpoint, 9 units in the x-direction and 7 units in the y-direction to point (21, 10).

### Example Question #1 : How To Find The Endpoints Of A Line Segment

The midpoint of a line segment is . If one endpoint of the line segment is , what is the other endpoint?

Explanation:

The midpoint formula can be used to solve this problem, where the midpoint is the average of the two coordinates.

We are given the midpoint and one endpoint. Plug these values into the formula.

Solve for the variables to find the coordinates of the second endpoint.

The final coordinates of the other endpoint are .

### Example Question #3 : How To Find The Endpoints Of A Line Segment

Suppose the midpoint of a line segment is  What are the endpoints of the segment?

Explanation:

The midpoint of a line segment is found using the formula .

The midpoint is given as  Going through the answer choices, only the points  and  yield the correct midpoint of .

### Example Question #2 : Midpoint Formula

What is the midpoint of the segment of

between  and ?

Explanation:

What is the midpoint of the segment of

between  and ?

To find this midpoint, you must first calculate the two end points.  Thus, substitute in for :

Then, for :

Thus, the two points in question are:

and

The midpoint of two points is:

Thus, for our data, this is:

or

### Example Question #3 : How To Find The Endpoints Of A Line Segment

If  is the midpoint of  and another point, what is that other point?

Explanation:

Recall that the midpoint's  and  values are the average of the  and  values of the two points in question.   Thus, if we call the other point , we know that:

and

Solve each equation accordingly:

For , multiply both sides by :

Thus,

The same goes for the other equation:

, so

Thus, our point is

### Example Question #4 : How To Find The Endpoints Of A Line Segment

If  is the midpoint of  and another point, what is that other point?

Explanation:

If  is the midpoint of  and another point, what is that other point?

Recall that the midpoint's  and  values are the average of the  and  values of the two points in question. Thus, if we call the other point , we know that:

and

Solve each equation accordingly:

For , multiply both sides by :

Thus,

The same goes for the other equation:

, so

Thus, our point is